Simple Complex Tori of Algebraic Dimension 0

Using Galois theory, we explicitly construct (in all complex dimensions $g\ge 2$) an infinite family of simple $g$-dimensional complex tori $T$ that enjoy the following properties:
$\bullet $ the Picard number of $T$ is $0;$ in particular, the algebraic dimension of $T$ is $0$;
$\bullet $ if $T^\vee $ is the dual of $T$, then $\mathrm {Hom}(T,T^\vee )=\{0\}$;
$\bullet $ the automorphism group $\mathrm {Aut}(T)$ of $T$ is isomorphic to $\{\pm 1\} \times \mathbb Z^{g-1}$;
$\bullet $ the endomorphism algebra $\mathrm {End}^0(T)$ of $T$ is a purely imaginary number field of degree $2g$.